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Probabilistic Modeling of Antibody Kinetics Post Infection and Vaccination: A Markov Chain Approach

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims to be the first to compute the probability distribution of population antibody levels under arbitrary sequences of infection and vaccination events, by coupling a time-inhomogeneous Markov chain of immune states with…

desk verdict A sincere multi-event extension of the authors' own Markov-chain antibody kinetics framework, with coherent math but in-sample validation and a real, acknowledged modeling inconsistency that needs fixing before the predictions can be taken at face value. read the letter →

arxiv 2507.10793 v2 pith:H6YJ7XUD submitted 2025-07-14 q-bio.PE math.PRphysics.bio-phq-bio.QMstat.ME

Rayanne A. Luke , Prajakta Bedekar , Lyndsey M. Muehling , Glenda Canderan , Yesun Lee , Wesley A. Cheng , Judith A. Woodfolk , Jeffrey M. Wilson
show 2 more authors
Pia S. Pannaraj Anthony J. Kearsley
This is my paper · ORCID
classification q-bio.PEmath.PRphysics.bio-phq-bio.QMstat.ME MSC 60J2062P1092D30
keywords antibodykineticstime-inhomogeneousMarkovchaingammadistributionimmuneeventsequencesSARS-CoV-2serologyvaccinationprevalenceestimationpersonaltrajectories
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that population antibody measurements can be interpreted as a weighted mixture of personal immune histories: a time-inhomogeneous Markov chain supplies the probability of each sequence of infections and vaccinations, and a gamma-distributed personal response model supplies the antibody distribution after that sequence. This is the first mathematical treatment the authors know of that simultaneously covers antibody levels, prevalence, multiple event classes, time dependence, and multiple immune events per person. If correct, a random blood draw can be assigned a probability under each personal trajectory, which supports serosurveillance, missed-event detection, and booster-timing decisions. The framework is demonstrated on two longitudinal SARS-CoV-2 cohorts with documented infections and vaccinations, plus population-transmission simulations.

What carries the argument

The machinery is a discrete-time, time-inhomogeneous Markov chain over immune states (naive, newly infected, newly vaccinated, previously infected, previously vaccinated) coupled to an additive gamma personal-response model. The transition matrix $S(T)$ gives the probability of moving between states at time step $T$, and products $H_T=S(T)S(T-1)\cdots S(0)$ enumerate every possible sequence of immune events. On the personal side, Eq. (7) defines the gamma shape after a sequence of events as a sum of peak-and-decay terms (one per event) plus the naive shape, with the scale fixed, so the response to multiple events is the convolution of independent event responses. The paper uses this pair of objects to write conditional probabilities for antibody measurements, then specializes to a time-homogeneous chain whose stationary distribution has a closed form.

What would settle it

Take longitudinal data in which a person's antibody level is measured immediately before a second infection or booster, then repeatedly afterwards, and test whether the size or variability of the post-event rise depends on the pre-event level. If high pre-event titers systematically produce smaller rises than the additive-shape model predicts, or if the spread visibly changes after repeated events, the central factorization fails; a likelihood-ratio comparison against a model with pre-event-titer-dependent shape and scale would settle it.

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Extended reading notes

Core claim

The central claim is that the population-level distribution of antibody measurements at a given calendar time $T$ is a convolution of transition probabilities and personal response densities: for any immune state, $P(r,T\mid X_T=I')$ equals a sum over possible event times of $R_I(r,T-t)$ multiplied by products of transition probabilities, and the same factorization holds for every multi-event trajectory. The multi-event personal response model is the load-bearing piece: after $M$ events the gamma shape parameter is the sum of event-specific terms, $\alpha_{M,\{z_m\}}(t,\tau)=\sum_{j=1}^{M-1}\theta_{z_j}\tau_j/(1+\phi_{z_j}\tau_j^{k_{z_j}})+\theta_{z_M}(t-\sum\tau_j)/(1+\phi_{z_M}(t-\sum\tau_j)^{k_{z_M}})+\alpha_N$, with the scale held at the naive value $\beta_N$. Because independent gamma variables with a common scale add to a gamma, each conditional probability factors cleanly into a sequence probability times a gamma density. The authors fit the parameters by maximum likelihood to two SARS-CoV-2 cohorts and report that most personal trajectories fall within model contours; a time-homogeneous version has a stationary distribution via the Perron-Frobenius theorem.

Load-bearing premise

Antibody responses to multiple immune events are assumed to combine as independent additive contributions to the gamma shape, with the spread fixed at the naive value; if a second event's response depends on the current antibody level or changes the variability of the response, the model's conditional probabilities no longer hold.

Editorial extensions

If this is right

  • Given incidence rates over time, the model outputs the full probability density of antibody levels at any calendar time, so a serosurvey sample can be compared with predicted population distributions rather than only mean prevalence.
  • Booster-timing decisions can be informed by the predicted fraction of the population below a chosen antibody threshold under alternative vaccination schedules.
  • An individual whose measured trajectory rises against the model's expected decay is flagged as a candidate for a missed infection or breakthrough event, as illustrated in the paper's outlier analyses.
  • Adding one more immune event to a personal history requires estimating only three new parameters, so long and varied event sequences remain tractable.
  • When incidence rates stabilize, the time-homogeneous chain has a stationary distribution that links long-run state prevalences to transition probabilities.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the additive-shape assumption in Eq. (7) could be tested directly with pre-event titer data; the paper does not run that test because its datasets document events but do not measure antibody level on the event day.
  • Editorial extension: interpret the conditional probabilities as posterior distributions over missed events: given a measurement and a documented event history, the model can rank hidden reinfections or breakthrough infections by probability. The paper's outlier-trajectory discussion points this way but does not formalize the inversion.
  • Editorial extension: the closed-form stationary distribution of the time-homogeneous chain suggests an inverse serosurveillance use: match observed cross-sectional antibody distributions to the model to estimate stabilized infection and vaccination incidence rates. The paper simulates forward transmission but does not solve this inverse problem.
  • Editorial extension: the same transition framework can be ported to influenza, RSV, or pertussis, where booster-timing questions could be recast as the first calendar time at which the predicted probability of falling below a protective antibody threshold crosses a chosen level.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a two-level modeling framework for population antibody kinetics. At the personal level, post-event antibody distributions are modeled as gamma densities whose shape parameter follows a peak-and-decline curve in time, extended to multiple immune events by adding shape contributions. At the population level, discrete-time Markov chains are used on a 13-state graph (two-event model, Section 4) and on a five-state graph (general time-inhomogeneous and time-homogeneous models, Section 5), yielding probabilities of immune-state trajectories that are convolved with personal response models to give conditional densities of antibody measurements on an absolute timeline. The framework is applied to two SARS-CoV-2 longitudinal datasets (Section 6) and to simulated population transmission (Section 7). The paper claims to be the first to simultaneously address antibody levels, prevalence, multiple immune-event classes, time dependence, and multiple immune events.

Significance. If the technical issues with the multi-event personal response model are resolved, the proposed framework would be a useful contribution, extending the authors' prior five-state Markov-chain antibody kinetics work to multiple immune events and providing explicitly enumerable conditional probabilities for the two-event case. The derivations in Sections 4.2 and 5.2 are careful applications of conditional probability and the law of total probability, and the paper is transparent about several of its limitations, including the non-vanishing asymptote of the multi-event shape (Eq. (37)) and the synthetic naive baseline used for Dataset 2. The graphical state models and closed-form conditional probabilities are likely to be usable by other researchers despite the empirical validation weaknesses.

major comments (3)
  1. [Section 3, Eqs. (5), (7); Section 8.5, Eq. (37)] The multi-event shape model evaluates each earlier event's gamma-shape contribution at the gap τ_j and freezes it for all later times t, rather than evaluating the single-event response at the total time elapsed since that event. This is inconsistent with the paper's own infinite-divisibility rationale in the paragraph following Eq. (3), which would give a combined shape of Σ_j [α1,zj(t_j) − αN] + αN with t_j = t − Σ_{i<j} τ_i. As written, Eq. (37) gives lim_{t→∞} α_n(t) = Σ_{i=1}^{n−1} θ_i τ_i/(1+φ_i τ_i^{k_i}) + αN > αN, so an individual with two events is predicted never to return to the naive baseline. The dismissal of this as a small discrepancy in Section 8.5 is unsupported; using the Dataset 1 infection parameters in Table 1, the frozen first-event contribution at τ = 100 days is about 11.9 excess shape units, whereas the single-event model would give about 3.0 at t = 400 days. Because every conditional probability in Sections 4.2 and 5.2 is built from these multi-event responses, this is a load-bearing assumption that needs either a corrected model or an explicit biological justification with quantitative sensitivity analysis.
  2. [Section 6, Tables 1-3, Table 2] The empirical support is weaker than claimed because the same data are used for fitting and evaluation. Parameters in Tables 1 and 3 are obtained by MLE from the same trajectories whose measurements are later scored for contour coverage and 'high probability' thresholds in Table 2. These statistics are in-sample goodness-of-fit measures, not predictive validation, and the high percentages (e.g., 96% for VV in Dataset 1) are expected to be optimistic. Please either reframe Section 6 and the abstract's 'validate' language as goodness of fit, or provide an out-of-sample evaluation such as cross-validation or a held-out cohort.
  3. [Section 6.2] The Dataset 2 analysis uses synthetic naive parameters α_n = 18.2 and β_n = 0.152, described as a 'reasonable guess' because no naive samples were collected. The single-event fits in Table 3 and the two-event VI fit, and therefore the Dataset 2 rows of Table 2, are all conditional on this unvalidated baseline. This should be presented as a substantial limitation rather than a minor detail, and a sensitivity analysis of the fitted parameters and coverage statistics to the naive parameters is needed to support the claimed validation.
minor comments (6)
  1. [Section 8.1, Eq. (36)] The state-count formula N_s = 4(2N_e − 1) + 1 gives 37 for N_e = 5, contradicting the sentence 'with 125 states necessary to model 5 possible immune events'; the intended formula is evidently 4(2^{N_e} − 1) + 1, which matches both the 13-state two-event case and the 125-state five-event example.
  2. [Section 4.2, Eq. (14)] The display 'Prob(X_T = I) = Prob(r, T|X_{T−1}=N, X_T=I) = R(r,0) = N(r)' equates a state probability with a conditional density; the left-hand side should be a density or the notation should distinguish the two.
  3. [Table 2] The table and surrounding text refer to 'probability > 0.001' and 'probability above 0.2'; these thresholds are density contour levels, not probabilities of an interval, and the text should say so to avoid confusion.
  4. [Appendix A, Eq. (A1)] The product 'S(4)S(3)S(2)S(2)S(1)S(0)' contains a repeated S(2); it should be S(4)S(3)S(2)S(1)S(0).
  5. [Section 2.1] The text states that time is discrete throughout, but the gamma response models in Eqs. (3)-(8) use a continuous time variable; the relationship between the discrete Markov chain time step and the continuous personal timeline should be stated explicitly.
  6. [Abstract and Section 1] The claim to be the first to simultaneously address topics (i)-(v) is difficult to verify; I suggest softening it or citing a broader comparison set.

Circularity Check

1 steps flagged · score 4.0 of 10

Core Markov-chain derivation is self-contained; Section 6's in-sample MLE fit is presented as validation and prediction, which is the main circularity.

  1. fitted input called prediction [Section 6 (Tables 1-3, Table 2) and Section 8.6]
    "To validate our models, we use combined clinical data from [23] and [22] and [24]... The results in Table 2 validate our modeling procedure, showing that a significant portion of the data lie within the contours... Our models predict the antibody response to sequences of infection and vaccination over time."

    The response parameters θ, φ, and k in Eqs. (3), (5), (7) and Tables 1/3 are obtained by MLE from the same Dataset 1/2 trajectories whose contour coverage is then reported in Table 2; for each two-event row, the first-event parameters are fixed from the single-event fit and the second-event parameters are fit to the second-event data. The Table 2 percentages therefore measure in-sample agreement between a fitted density and its own training data, and the paper's language 'validate'/'predict' presents this fitted input as an independent predictive success. The model could still be useful, but this particular numerical demonstration does not provide out-of-sample evidence, so the predictive claim is, at this point, a restatement of the fit rather than a separate result.

full rationale

The mathematical derivation chain is largely self-contained and non-circular. The conditional probabilities in Sections 4.2 and 5.2 are obtained from the definition of conditional probability, the law of total probability, and products of the Markov transition matrices, with the personal-response gamma densities entering as explicitly stated modeling choices. No step equates a derived quantity to its own definition, and no 'uniqueness' result is imported from the authors' prior work to force the choice of response model; Eq. (3) is introduced as 'a modeling choice.' The main circularity is confined to the numerical demonstration: Section 6 fits all shape/scale parameters by MLE on the same trajectories later scored in Table 2, so the reported contour coverage is in-sample, and Section 8.6's claim that the model 'predicts' antibody response converts this fitted input into a predictive statement. The paper itself flags a related limitation in Section 8.5 (Eq. (37), multi-event shape no longer returns to αN), but that is an internal modeling inconsistency rather than a circularity. Prior self-citations ([20], [21]) supply the single-event blueprint but are not used as an unverified authority for the multi-event result; the multi-event Markov formulation has independent mathematical content. Overall, the central derivation is not forced by self-citation or by definition, but the in-sample validation inflates the predictive claim, giving a partial circularity score of 4.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The model rests on a small number of fitted gamma parameters and a set of explicit biological and structural assumptions. The personal response model contributes three parameters per event type, plus the naive shape and scale; all are fitted to the same data used for validation. The Markov chain itself adds no free parameters, since transition probabilities are defined from incidence rates, which are inputs. The assumptions of additive independent shape contributions and fixed scale, the initial all-naive condition, and the two-event restriction are stated but not independently tested.

free parameters (7)
  • Naive gamma shape αN = 1.23 (Dataset 1); 18.2 (synthetic, Dataset 2)
    Fitted via MLE for Dataset 1; hand-picked for Dataset 2 because no naive samples were available. Central to the personal response model since it is the baseline to which event contributions are added.
  • Naive gamma scale βN = 0.256 (Dataset 1); 0.152 (synthetic, Dataset 2)
    Same as above; also assumed unchanged for all event responses.
  • Single infection response parameters θI1, φI1, kI1 = θ=79.0, φ=5.63, k=2.10 (Dataset 1); θ=2757, φ=66.8, k=1.04 (Dataset 2)
    MLE fit to single-infection antibody data. Determine the magnitude, timing, and decay of the infection response.
  • Single vaccination response parameters θV1, φV1, kV1 = θ=520, φ=51.4, k=1.51 (Dataset 1); θ=8113, φ=135, k=1.11 (Dataset 2)
    MLE fit to vaccination data.
  • Second-event parameters θV2, φV2, kV2 (booster after vaccination) = θ=106, φ=11.1, k=1.70 (Dataset 1)
    MLE fit to booster data; the first-event parameters are held fixed from the single-event fit.
  • Second-event parameters θV2, φV2, kV2 (vaccination after infection) = θ=101, φ=14.3, k=1.93 (Dataset 1); θ=790, φ=29.8, k=1.26 (Dataset 2)
    MLE fit to VI data; first-event parameters are taken from the single-infection fit.
  • Validation thresholds = probability > 0.001 contour; probability > 0.2 high-probability
    Arbitrary thresholds chosen post hoc to summarize goodness of fit; they affect the reported validation percentages in Table 2.
assumptions (6)
  • domain assumption Antibody response to an immune event follows a gamma distribution with shape α(t) = θt/(1+φt^k) + αN and fixed scale βN
    Section 3. This functional form is a modeling choice intended to reproduce rise, peak, and wane. It is not derived from biology.
  • domain assumption Responses to successive immune events are independent and additive in the gamma shape parameter, with a common fixed scale βN
    Section 3, Eqs. (5)-(7). The entire multi-event personal response model depends on this. The paper states the independence assumption without validation.
  • domain assumption The population begins entirely in the naive state at disease emergence
    Section 4.1: X1 = e1. This is used to set initial conditions for all prevalence calculations.
  • domain assumption Markov property: transition probabilities at the next time step depend only on current state
    Section 2.1: 'given a current state and time t in personal timeline, one can compute the transition probability for the next time step'.
  • domain assumption In the two-event model, no return to naive is allowed after an immune event, and at most two events can occur
    Section 4: 'we also forbid explicit movement back to the naive state... assumes that the time scale of the problem is short.' Section 5.2 sets sNI'=sNV'=0 for the conditional probability derivation.
  • standard math Perron-Frobenius theorem applies to the time-homogeneous transition matrix
    Section 5.3: used to establish unique stationary distribution, requiring irreducibility and aperiodicity, which the authors assert holds when all transitions are positive.

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Cite this review

Pith. "Pith review of Probabilistic Modeling of Antibody Kinetics Post Infection and Vaccination: A Markov Chain Approach." pith.science (2026). https://pith.science/paper/H6YJ7XUD

@misc{pith2026250710793,
  author       = {Pith},
  title        = {Pith review of: Probabilistic Modeling of Antibody Kinetics Post Infection and Vaccination: A Markov Chain Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6YJ7XUD}},
  note         = {Machine review of arXiv:2507.10793}
}
read the original abstract

Understanding the dynamics of antibody levels is crucial for characterizing the time-dependent response to immune events: either infections or vaccinations. The sequence and timing of these events significantly influence antibody level changes. Despite extensive interest in the topic in the recent years and many experimental studies, the effect of immune event sequences on antibody levels is not well understood. Moreover, disease or vaccination prevalence in the population are time-dependent. This, alongside the complexities of personal antibody kinetics, makes it difficult to analyze a sample immune measurement from a population. As a solution, we design a rigorous mathematical characterization in terms of a time-inhomogeneous Markov chain model for event-to-event transitions coupled with a probabilistic framework for the post-event antibody kinetics of multiple immune events. We demonstrate that this is an ideal model for immune event sequences, referred to as personal trajectories. This novel modeling framework surpasses the susceptible-infected-recovered (SIR) characterizations by rigorously tracking the probability distribution of population antibody response across time. To illustrate our ideas, we apply our mathematical framework to longitudinal severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) data from individuals with multiple documented infection and vaccination events. Our work is an important step towards a comprehensive understanding of antibody kinetics that could lead to an effective way to analyze the protective power of natural immunity or vaccination, predict missed immune events at an individual level, and inform booster timing recommendations.

Figures

Figures reproduced from arXiv: 2507.10793 by the authors.

Figure 1
Figure 1. Graph of a model where two events are allowable: two infections, two vaccinations, infection after [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 1
Figure 1. Since the set of previously (once) infected individuals can be partitioned by the day on which they [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. We let sN (T) denote the weight of the degenerate edge to N from N, or the probability of staying na¨ıve. The probabilities sIN (T) and sVN (T) weight the edges to I from N and to V from N, indicating infection or vaccination, respectively; the other transition probabilities have analogous interpretations. The only disallowed transitions are those that go from newly infected to any state other than previously infect… view at source ↗
Figures from the paper (11 more)
Figure 3
Figure 3. Figure 3: Log-transformed antibody measurements from Dataset 1 with corresponding probability models [PITH_FULL_IMAGE:figures/full_fig_p014_3.png]
Figure 4
Figure 4. Figure 4: Non-titration-extrapolated, log-transformed antibody measurements (markers) from the infected [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: For Dataset 2: (a) Linear fit to find intercept to translate our calculated AUC to that reported by [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Log-transformed antibody measurements from the infected (I), vaccinated (V), and infected then [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Log-transformed antibody measurements from the na¨ıve (synthetic), infected (I), vaccinated (V), [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Simulations for a population with 1000 individuals with a time-homogeneous transition matrix [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Simulations for a population with 1000 individuals with a time-homogeneous transition matrix [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Log-transformed antibody measurements from the infected (I) (a),(b) and vaccinated (V) (c),(d) [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Log-transformed antibody measurements from the vaccinated then boosted (VV) population from [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Log-transformed antibody measurements from the infected, then vaccinated (VI) population from [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Schematic summarizing the main mathematical ideas in this paper. [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.